Graphics.Rendering.Chart.Axis.Types:linMap from Chart-1.5.3

Percentage Accurate: 67.7% → 87.9%

Time: 6.9s

Alternatives: 12

Speedup: 0.8×

Specification

Math

\[\begin{array}{l} \\ x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 (+ x (/ (* (- y x) (- z t)) (- a t))))

C

double code(double x, double y, double z, double t, double a) {
	return x + (((y - x) * (z - t)) / (a - t));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = x + (((y - x) * (z - t)) / (a - t))
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return x + (((y - x) * (z - t)) / (a - t));
}

Python

def code(x, y, z, t, a):
	return x + (((y - x) * (z - t)) / (a - t))

Julia

function code(x, y, z, t, a)
	return Float64(x + Float64(Float64(Float64(y - x) * Float64(z - t)) / Float64(a - t)))
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = x + (((y - x) * (z - t)) / (a - t));
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(x + N[(N[(N[(y - x), $MachinePrecision] * N[(z - t), $MachinePrecision]), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 67.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 (+ x (/ (* (- y x) (- z t)) (- a t))))

C

double code(double x, double y, double z, double t, double a) {
	return x + (((y - x) * (z - t)) / (a - t));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = x + (((y - x) * (z - t)) / (a - t))
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return x + (((y - x) * (z - t)) / (a - t));
}

Python

def code(x, y, z, t, a):
	return x + (((y - x) * (z - t)) / (a - t))

Julia

function code(x, y, z, t, a)
	return Float64(x + Float64(Float64(Float64(y - x) * Float64(z - t)) / Float64(a - t)))
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = x + (((y - x) * (z - t)) / (a - t));
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(x + N[(N[(N[(y - x), $MachinePrecision] * N[(z - t), $MachinePrecision]), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 87.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y - x, \frac{a - z}{t}, y\right)\\ \mathbf{if}\;t \leq -1 \cdot 10^{+160}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 3.75 \cdot 10^{+73}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - z}{t - a}, y - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (- y x) (/ (- a z) t) y)))
   (if (<= t -1e+160)
     t_1
     (if (<= t 3.75e+73) (fma (/ (- t z) (- t a)) (- y x) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((y - x), ((a - z) / t), y);
	double tmp;
	if (t <= -1e+160) {
		tmp = t_1;
	} else if (t <= 3.75e+73) {
		tmp = fma(((t - z) / (t - a)), (y - x), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(y - x), Float64(Float64(a - z) / t), y)
	tmp = 0.0
	if (t <= -1e+160)
		tmp = t_1;
	elseif (t <= 3.75e+73)
		tmp = fma(Float64(Float64(t - z) / Float64(t - a)), Float64(y - x), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y - x), $MachinePrecision] * N[(N[(a - z), $MachinePrecision] / t), $MachinePrecision] + y), $MachinePrecision]}, If[LessEqual[t, -1e+160], t$95$1, If[LessEqual[t, 3.75e+73], N[(N[(N[(t - z), $MachinePrecision] / N[(t - a), $MachinePrecision]), $MachinePrecision] * N[(y - x), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -1.00000000000000001e160 or 3.75e73 < t

   1. Initial program 67.7%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]

   2. Taylor expanded in t around -inf

      \[\leadsto \color{blue}{y + -1 \cdot \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}} \]
   3. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto y + \color{blue}{-1 \cdot \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}} \]

      2. lower-*.f64 N/A

         \[\leadsto y + -1 \cdot \color{blue}{\frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}} \]

      3. lower-/.f64 N/A

         \[\leadsto y + -1 \cdot \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{\color{blue}{t}} \]

      4. lower--.f64 N/A

         \[\leadsto y + -1 \cdot \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t} \]

      5. lower-*.f64 N/A

         \[\leadsto y + -1 \cdot \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t} \]

      6. lower--.f64 N/A

         \[\leadsto y + -1 \cdot \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t} \]

      7. lower-*.f64 N/A

         \[\leadsto y + -1 \cdot \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t} \]

      8. lower--.f64 46.0

         \[\leadsto y + -1 \cdot \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t} \]

   4. Applied rewrites 46.0%

      \[\leadsto \color{blue}{y + -1 \cdot \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto y + \color{blue}{-1 \cdot \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}} \]

      2. +-commutative N/A

         \[\leadsto -1 \cdot \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t} + \color{blue}{y} \]

      3. add-flip N/A

         \[\leadsto -1 \cdot \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t} - \color{blue}{\left(\mathsf{neg}\left(y\right)\right)} \]

      4. sub-flip N/A

         \[\leadsto -1 \cdot \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t} + \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(y\right)\right)\right)\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto -1 \cdot \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t} + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}\right)\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right)\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{y}\right)\right)\right)\right) \]

      8. distribute-neg-frac N/A

         \[\leadsto \frac{\mathsf{neg}\left(\left(z \cdot \left(y - x\right) - a \cdot \left(y - x\right)\right)\right)}{t} + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right)\right) \]

      9. lift--.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\left(z \cdot \left(y - x\right) - a \cdot \left(y - x\right)\right)\right)}{t} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(y\right)\right)\right)\right) \]

      10. sub-negate-rev N/A

          \[\leadsto \frac{a \cdot \left(y - x\right) - z \cdot \left(y - x\right)}{t} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{y}\right)\right)\right)\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{a \cdot \left(y - x\right) - z \cdot \left(y - x\right)}{t} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(y\right)\right)\right)\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{a \cdot \left(y - x\right) - z \cdot \left(y - x\right)}{t} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(y\right)\right)\right)\right) \]

      13. distribute-rgt-out-- N/A

          \[\leadsto \frac{\left(y - x\right) \cdot \left(a - z\right)}{t} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{y}\right)\right)\right)\right) \]

      14. associate-/l* N/A

          \[\leadsto \left(y - x\right) \cdot \frac{a - z}{t} + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right)\right) \]

      15. remove-double-neg N/A

          \[\leadsto \left(y - x\right) \cdot \frac{a - z}{t} + y \]

      16. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(y - x, \color{blue}{\frac{a - z}{t}}, y\right) \]

   6. Applied rewrites 53.2%

      \[\leadsto \mathsf{fma}\left(y - x, \color{blue}{\frac{a - z}{t}}, y\right) \]

   if -1.00000000000000001e160 < t < 3.75e73

   1. Initial program 67.7%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} + x} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}} + x \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(y - x\right) \cdot \left(z - t\right)}}{a - t} + x \]

      5. associate-/l* N/A

         \[\leadsto \color{blue}{\left(y - x\right) \cdot \frac{z - t}{a - t}} + x \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\frac{z - t}{a - t} \cdot \left(y - x\right)} + x \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z - t}{a - t}, y - x, x\right)} \]

      8. frac-2neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\mathsf{neg}\left(\left(z - t\right)\right)}{\mathsf{neg}\left(\left(a - t\right)\right)}}, y - x, x\right) \]

      9. lift--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\mathsf{neg}\left(\color{blue}{\left(z - t\right)}\right)}{\mathsf{neg}\left(\left(a - t\right)\right)}, y - x, x\right) \]

      10. sub-negate-rev N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{t - z}}{\mathsf{neg}\left(\left(a - t\right)\right)}, y - x, x\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{t - z}{\mathsf{neg}\left(\left(a - t\right)\right)}}, y - x, x\right) \]

      12. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{t - z}}{\mathsf{neg}\left(\left(a - t\right)\right)}, y - x, x\right) \]

      13. lift--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{t - z}{\mathsf{neg}\left(\color{blue}{\left(a - t\right)}\right)}, y - x, x\right) \]

      14. sub-negate-rev N/A

          \[\leadsto \mathsf{fma}\left(\frac{t - z}{\color{blue}{t - a}}, y - x, x\right) \]

      15. lower--.f64 83.5

          \[\leadsto \mathsf{fma}\left(\frac{t - z}{\color{blue}{t - a}}, y - x, x\right) \]

   3. Applied rewrites 83.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t - z}{t - a}, y - x, x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 76.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y - x, \frac{a - z}{t}, y\right)\\ \mathbf{if}\;t \leq -6.5 \cdot 10^{+73}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 3.75 \cdot 10^{+73}:\\ \;\;\;\;x - \frac{\left(y - x\right) \cdot z}{t - a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (- y x) (/ (- a z) t) y)))
   (if (<= t -6.5e+73)
     t_1
     (if (<= t 3.75e+73) (- x (/ (* (- y x) z) (- t a))) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((y - x), ((a - z) / t), y);
	double tmp;
	if (t <= -6.5e+73) {
		tmp = t_1;
	} else if (t <= 3.75e+73) {
		tmp = x - (((y - x) * z) / (t - a));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(y - x), Float64(Float64(a - z) / t), y)
	tmp = 0.0
	if (t <= -6.5e+73)
		tmp = t_1;
	elseif (t <= 3.75e+73)
		tmp = Float64(x - Float64(Float64(Float64(y - x) * z) / Float64(t - a)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y - x), $MachinePrecision] * N[(N[(a - z), $MachinePrecision] / t), $MachinePrecision] + y), $MachinePrecision]}, If[LessEqual[t, -6.5e+73], t$95$1, If[LessEqual[t, 3.75e+73], N[(x - N[(N[(N[(y - x), $MachinePrecision] * z), $MachinePrecision] / N[(t - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -6.5000000000000001e73 or 3.75e73 < t

   1. Initial program 67.7%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]

   2. Taylor expanded in t around -inf

      \[\leadsto \color{blue}{y + -1 \cdot \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}} \]
   3. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto y + \color{blue}{-1 \cdot \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}} \]

      2. lower-*.f64 N/A

         \[\leadsto y + -1 \cdot \color{blue}{\frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}} \]

      3. lower-/.f64 N/A

         \[\leadsto y + -1 \cdot \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{\color{blue}{t}} \]

      4. lower--.f64 N/A

         \[\leadsto y + -1 \cdot \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t} \]

      5. lower-*.f64 N/A

         \[\leadsto y + -1 \cdot \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t} \]

      6. lower--.f64 N/A

         \[\leadsto y + -1 \cdot \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t} \]

      7. lower-*.f64 N/A

         \[\leadsto y + -1 \cdot \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t} \]

      8. lower--.f64 46.0

         \[\leadsto y + -1 \cdot \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t} \]

   4. Applied rewrites 46.0%

      \[\leadsto \color{blue}{y + -1 \cdot \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto y + \color{blue}{-1 \cdot \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}} \]

      2. +-commutative N/A

         \[\leadsto -1 \cdot \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t} + \color{blue}{y} \]

      3. add-flip N/A

         \[\leadsto -1 \cdot \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t} - \color{blue}{\left(\mathsf{neg}\left(y\right)\right)} \]

      4. sub-flip N/A

         \[\leadsto -1 \cdot \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t} + \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(y\right)\right)\right)\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto -1 \cdot \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t} + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}\right)\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right)\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{y}\right)\right)\right)\right) \]

      8. distribute-neg-frac N/A

         \[\leadsto \frac{\mathsf{neg}\left(\left(z \cdot \left(y - x\right) - a \cdot \left(y - x\right)\right)\right)}{t} + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right)\right) \]

      9. lift--.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\left(z \cdot \left(y - x\right) - a \cdot \left(y - x\right)\right)\right)}{t} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(y\right)\right)\right)\right) \]

      10. sub-negate-rev N/A

          \[\leadsto \frac{a \cdot \left(y - x\right) - z \cdot \left(y - x\right)}{t} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{y}\right)\right)\right)\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{a \cdot \left(y - x\right) - z \cdot \left(y - x\right)}{t} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(y\right)\right)\right)\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{a \cdot \left(y - x\right) - z \cdot \left(y - x\right)}{t} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(y\right)\right)\right)\right) \]

      13. distribute-rgt-out-- N/A

          \[\leadsto \frac{\left(y - x\right) \cdot \left(a - z\right)}{t} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{y}\right)\right)\right)\right) \]

      14. associate-/l* N/A

          \[\leadsto \left(y - x\right) \cdot \frac{a - z}{t} + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right)\right) \]

      15. remove-double-neg N/A

          \[\leadsto \left(y - x\right) \cdot \frac{a - z}{t} + y \]

      16. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(y - x, \color{blue}{\frac{a - z}{t}}, y\right) \]

   6. Applied rewrites 53.2%

      \[\leadsto \mathsf{fma}\left(y - x, \color{blue}{\frac{a - z}{t}}, y\right) \]

   if -6.5000000000000001e73 < t < 3.75e73

   1. Initial program 67.7%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]

   2. Taylor expanded in z around inf

      \[\leadsto x + \frac{\color{blue}{z \cdot \left(y - x\right)}}{a - t} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto x + \frac{z \cdot \color{blue}{\left(y - x\right)}}{a - t} \]

      2. lower--.f64 54.0

         \[\leadsto x + \frac{z \cdot \left(y - \color{blue}{x}\right)}{a - t} \]

   4. Applied rewrites 54.0%

      \[\leadsto x + \frac{\color{blue}{z \cdot \left(y - x\right)}}{a - t} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \frac{z \cdot \left(y - x\right)}{a - t}} \]

      2. add-flip N/A

         \[\leadsto \color{blue}{x - \left(\mathsf{neg}\left(\frac{z \cdot \left(y - x\right)}{a - t}\right)\right)} \]

      3. lower--.f64 N/A

         \[\leadsto \color{blue}{x - \left(\mathsf{neg}\left(\frac{z \cdot \left(y - x\right)}{a - t}\right)\right)} \]

      4. lift-/.f64 N/A

         \[\leadsto x - \left(\mathsf{neg}\left(\color{blue}{\frac{z \cdot \left(y - x\right)}{a - t}}\right)\right) \]

      5. distribute-neg-frac2 N/A

         \[\leadsto x - \color{blue}{\frac{z \cdot \left(y - x\right)}{\mathsf{neg}\left(\left(a - t\right)\right)}} \]

      6. lower-/.f64 N/A

         \[\leadsto x - \color{blue}{\frac{z \cdot \left(y - x\right)}{\mathsf{neg}\left(\left(a - t\right)\right)}} \]

      7. lift-*.f64 N/A

         \[\leadsto x - \frac{z \cdot \color{blue}{\left(y - x\right)}}{\mathsf{neg}\left(\left(a - t\right)\right)} \]

      8. *-commutative N/A

         \[\leadsto x - \frac{\left(y - x\right) \cdot \color{blue}{z}}{\mathsf{neg}\left(\left(a - t\right)\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto x - \frac{\left(y - x\right) \cdot \color{blue}{z}}{\mathsf{neg}\left(\left(a - t\right)\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto x - \frac{\left(y - x\right) \cdot \color{blue}{z}}{\mathsf{neg}\left(\mathsf{Rewrite=>}\left(lift--.f64, \left(a - t\right)\right)\right)} \]

      11. lift-*.f64 N/A

          \[\leadsto x - \frac{\left(y - x\right) \cdot \color{blue}{z}}{\mathsf{Rewrite<=}\left(sub-negate-rev, \left(t - a\right)\right)} \]

      12. lift-*.f64 N/A

          \[\leadsto x - \frac{\left(y - x\right) \cdot \color{blue}{z}}{\mathsf{Rewrite=>}\left(lower--.f64, \left(t - a\right)\right)} \]

   6. Applied rewrites 54.0%

      \[\leadsto \color{blue}{x - \frac{\left(y - x\right) \cdot z}{t - a}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 71.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.85 \cdot 10^{-19}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{t - a}, y - x, x\right)\\ \mathbf{elif}\;a \leq -1.18 \cdot 10^{-82}:\\ \;\;\;\;\frac{z}{t - a} \cdot \left(x - y\right)\\ \mathbf{elif}\;a \leq 6000000:\\ \;\;\;\;\mathsf{fma}\left(y - x, \frac{a - z}{t}, y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, y - x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -2.85e-19)
   (fma (/ t (- t a)) (- y x) x)
   (if (<= a -1.18e-82)
     (* (/ z (- t a)) (- x y))
     (if (<= a 6000000.0)
       (fma (- y x) (/ (- a z) t) y)
       (fma (/ z a) (- y x) x)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -2.85e-19) {
		tmp = fma((t / (t - a)), (y - x), x);
	} else if (a <= -1.18e-82) {
		tmp = (z / (t - a)) * (x - y);
	} else if (a <= 6000000.0) {
		tmp = fma((y - x), ((a - z) / t), y);
	} else {
		tmp = fma((z / a), (y - x), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -2.85e-19)
		tmp = fma(Float64(t / Float64(t - a)), Float64(y - x), x);
	elseif (a <= -1.18e-82)
		tmp = Float64(Float64(z / Float64(t - a)) * Float64(x - y));
	elseif (a <= 6000000.0)
		tmp = fma(Float64(y - x), Float64(Float64(a - z) / t), y);
	else
		tmp = fma(Float64(z / a), Float64(y - x), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -2.85e-19], N[(N[(t / N[(t - a), $MachinePrecision]), $MachinePrecision] * N[(y - x), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[a, -1.18e-82], N[(N[(z / N[(t - a), $MachinePrecision]), $MachinePrecision] * N[(x - y), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 6000000.0], N[(N[(y - x), $MachinePrecision] * N[(N[(a - z), $MachinePrecision] / t), $MachinePrecision] + y), $MachinePrecision], N[(N[(z / a), $MachinePrecision] * N[(y - x), $MachinePrecision] + x), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if a < -2.84999999999999976e-19

   1. Initial program 67.7%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} + x} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}} + x \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(y - x\right) \cdot \left(z - t\right)}}{a - t} + x \]

      5. associate-/l* N/A

         \[\leadsto \color{blue}{\left(y - x\right) \cdot \frac{z - t}{a - t}} + x \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\frac{z - t}{a - t} \cdot \left(y - x\right)} + x \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z - t}{a - t}, y - x, x\right)} \]

      8. frac-2neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\mathsf{neg}\left(\left(z - t\right)\right)}{\mathsf{neg}\left(\left(a - t\right)\right)}}, y - x, x\right) \]

      9. lift--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\mathsf{neg}\left(\color{blue}{\left(z - t\right)}\right)}{\mathsf{neg}\left(\left(a - t\right)\right)}, y - x, x\right) \]

      10. sub-negate-rev N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{t - z}}{\mathsf{neg}\left(\left(a - t\right)\right)}, y - x, x\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{t - z}{\mathsf{neg}\left(\left(a - t\right)\right)}}, y - x, x\right) \]

      12. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{t - z}}{\mathsf{neg}\left(\left(a - t\right)\right)}, y - x, x\right) \]

      13. lift--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{t - z}{\mathsf{neg}\left(\color{blue}{\left(a - t\right)}\right)}, y - x, x\right) \]

      14. sub-negate-rev N/A

          \[\leadsto \mathsf{fma}\left(\frac{t - z}{\color{blue}{t - a}}, y - x, x\right) \]

      15. lower--.f64 83.5

          \[\leadsto \mathsf{fma}\left(\frac{t - z}{\color{blue}{t - a}}, y - x, x\right) \]

   3. Applied rewrites 83.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t - z}{t - a}, y - x, x\right)} \]

   4. Taylor expanded in z around 0

      \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{t}}{t - a}, y - x, x\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 46.6%

      \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{t}}{t - a}, y - x, x\right) \]

   if -2.84999999999999976e-19 < a < -1.1799999999999999e-82

   1. Initial program 67.7%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]

   2. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(\frac{y}{a - t} - \frac{x}{a - t}\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto z \cdot \color{blue}{\left(\frac{y}{a - t} - \frac{x}{a - t}\right)} \]

      2. lower--.f64 N/A

         \[\leadsto z \cdot \left(\frac{y}{a - t} - \color{blue}{\frac{x}{a - t}}\right) \]

      3. lower-/.f64 N/A

         \[\leadsto z \cdot \left(\frac{y}{a - t} - \frac{\color{blue}{x}}{a - t}\right) \]

      4. lower--.f64 N/A

         \[\leadsto z \cdot \left(\frac{y}{a - t} - \frac{x}{a - t}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto z \cdot \left(\frac{y}{a - t} - \frac{x}{\color{blue}{a - t}}\right) \]

      6. lower--.f64 41.7

         \[\leadsto z \cdot \left(\frac{y}{a - t} - \frac{x}{a - \color{blue}{t}}\right) \]

   4. Applied rewrites 41.7%

      \[\leadsto \color{blue}{z \cdot \left(\frac{y}{a - t} - \frac{x}{a - t}\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto z \cdot \color{blue}{\left(\frac{y}{a - t} - \frac{x}{a - t}\right)} \]

      2. lift--.f64 N/A

         \[\leadsto z \cdot \left(\frac{y}{a - t} - \color{blue}{\frac{x}{a - t}}\right) \]

      3. lift-/.f64 N/A

         \[\leadsto z \cdot \left(\frac{y}{a - t} - \frac{\color{blue}{x}}{a - t}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto z \cdot \left(\frac{y}{a - t} - \frac{x}{\color{blue}{a - t}}\right) \]

      5. frac-sub N/A

         \[\leadsto z \cdot \frac{y \cdot \left(a - t\right) - \left(a - t\right) \cdot x}{\color{blue}{\left(a - t\right) \cdot \left(a - t\right)}} \]

      6. associate-*r/ N/A

         \[\leadsto \frac{z \cdot \left(y \cdot \left(a - t\right) - \left(a - t\right) \cdot x\right)}{\color{blue}{\left(a - t\right) \cdot \left(a - t\right)}} \]

      7. sqr-neg-rev N/A

         \[\leadsto \frac{z \cdot \left(y \cdot \left(a - t\right) - \left(a - t\right) \cdot x\right)}{\left(\mathsf{neg}\left(\left(a - t\right)\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\left(a - t\right)\right)\right)}} \]

      8. lift--.f64 N/A

         \[\leadsto \frac{z \cdot \left(y \cdot \left(a - t\right) - \left(a - t\right) \cdot x\right)}{\left(\mathsf{neg}\left(\left(a - t\right)\right)\right) \cdot \left(\mathsf{neg}\left(\left(\color{blue}{a} - t\right)\right)\right)} \]

      9. sub-negate-rev N/A

         \[\leadsto \frac{z \cdot \left(y \cdot \left(a - t\right) - \left(a - t\right) \cdot x\right)}{\left(t - a\right) \cdot \left(\mathsf{neg}\left(\color{blue}{\left(a - t\right)}\right)\right)} \]

      10. lift--.f64 N/A

          \[\leadsto \frac{z \cdot \left(y \cdot \left(a - t\right) - \left(a - t\right) \cdot x\right)}{\left(t - a\right) \cdot \left(\mathsf{neg}\left(\color{blue}{\left(a - t\right)}\right)\right)} \]

      11. lift--.f64 N/A

          \[\leadsto \frac{z \cdot \left(y \cdot \left(a - t\right) - \left(a - t\right) \cdot x\right)}{\left(t - a\right) \cdot \left(\mathsf{neg}\left(\left(a - t\right)\right)\right)} \]

      12. sub-negate-rev N/A

          \[\leadsto \frac{z \cdot \left(y \cdot \left(a - t\right) - \left(a - t\right) \cdot x\right)}{\left(t - a\right) \cdot \left(t - \color{blue}{a}\right)} \]

      13. lift--.f64 N/A

          \[\leadsto \frac{z \cdot \left(y \cdot \left(a - t\right) - \left(a - t\right) \cdot x\right)}{\left(t - a\right) \cdot \left(t - \color{blue}{a}\right)} \]

      14. times-frac N/A

          \[\leadsto \frac{z}{t - a} \cdot \color{blue}{\frac{y \cdot \left(a - t\right) - \left(a - t\right) \cdot x}{t - a}} \]

      15. frac-2neg-rev N/A

          \[\leadsto \frac{z}{t - a} \cdot \frac{\mathsf{neg}\left(\left(y \cdot \left(a - t\right) - \left(a - t\right) \cdot x\right)\right)}{\color{blue}{\mathsf{neg}\left(\left(t - a\right)\right)}} \]

   6. Applied rewrites 43.2%

      \[\leadsto \frac{z}{t - a} \cdot \color{blue}{\left(x - y\right)} \]

   if -1.1799999999999999e-82 < a < 6e6

   1. Initial program 67.7%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]

   2. Taylor expanded in t around -inf

      \[\leadsto \color{blue}{y + -1 \cdot \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}} \]
   3. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto y + \color{blue}{-1 \cdot \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}} \]

      2. lower-*.f64 N/A

         \[\leadsto y + -1 \cdot \color{blue}{\frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}} \]

      3. lower-/.f64 N/A

         \[\leadsto y + -1 \cdot \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{\color{blue}{t}} \]

      4. lower--.f64 N/A

         \[\leadsto y + -1 \cdot \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t} \]

      5. lower-*.f64 N/A

         \[\leadsto y + -1 \cdot \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t} \]

      6. lower--.f64 N/A

         \[\leadsto y + -1 \cdot \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t} \]

      7. lower-*.f64 N/A

         \[\leadsto y + -1 \cdot \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t} \]

      8. lower--.f64 46.0

         \[\leadsto y + -1 \cdot \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t} \]

   4. Applied rewrites 46.0%

      \[\leadsto \color{blue}{y + -1 \cdot \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto y + \color{blue}{-1 \cdot \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}} \]

      2. +-commutative N/A

         \[\leadsto -1 \cdot \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t} + \color{blue}{y} \]

      3. add-flip N/A

         \[\leadsto -1 \cdot \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t} - \color{blue}{\left(\mathsf{neg}\left(y\right)\right)} \]

      4. sub-flip N/A

         \[\leadsto -1 \cdot \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t} + \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(y\right)\right)\right)\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto -1 \cdot \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t} + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}\right)\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right)\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{y}\right)\right)\right)\right) \]

      8. distribute-neg-frac N/A

         \[\leadsto \frac{\mathsf{neg}\left(\left(z \cdot \left(y - x\right) - a \cdot \left(y - x\right)\right)\right)}{t} + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right)\right) \]

      9. lift--.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\left(z \cdot \left(y - x\right) - a \cdot \left(y - x\right)\right)\right)}{t} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(y\right)\right)\right)\right) \]

      10. sub-negate-rev N/A

          \[\leadsto \frac{a \cdot \left(y - x\right) - z \cdot \left(y - x\right)}{t} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{y}\right)\right)\right)\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{a \cdot \left(y - x\right) - z \cdot \left(y - x\right)}{t} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(y\right)\right)\right)\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{a \cdot \left(y - x\right) - z \cdot \left(y - x\right)}{t} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(y\right)\right)\right)\right) \]

      13. distribute-rgt-out-- N/A

          \[\leadsto \frac{\left(y - x\right) \cdot \left(a - z\right)}{t} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{y}\right)\right)\right)\right) \]

      14. associate-/l* N/A

          \[\leadsto \left(y - x\right) \cdot \frac{a - z}{t} + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right)\right) \]

      15. remove-double-neg N/A

          \[\leadsto \left(y - x\right) \cdot \frac{a - z}{t} + y \]

      16. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(y - x, \color{blue}{\frac{a - z}{t}}, y\right) \]

   6. Applied rewrites 53.2%

      \[\leadsto \mathsf{fma}\left(y - x, \color{blue}{\frac{a - z}{t}}, y\right) \]

   if 6e6 < a

   1. Initial program 67.7%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} + x} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}} + x \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(y - x\right) \cdot \left(z - t\right)}}{a - t} + x \]

      5. associate-/l* N/A

         \[\leadsto \color{blue}{\left(y - x\right) \cdot \frac{z - t}{a - t}} + x \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\frac{z - t}{a - t} \cdot \left(y - x\right)} + x \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z - t}{a - t}, y - x, x\right)} \]

      8. frac-2neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\mathsf{neg}\left(\left(z - t\right)\right)}{\mathsf{neg}\left(\left(a - t\right)\right)}}, y - x, x\right) \]

      9. lift--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\mathsf{neg}\left(\color{blue}{\left(z - t\right)}\right)}{\mathsf{neg}\left(\left(a - t\right)\right)}, y - x, x\right) \]

      10. sub-negate-rev N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{t - z}}{\mathsf{neg}\left(\left(a - t\right)\right)}, y - x, x\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{t - z}{\mathsf{neg}\left(\left(a - t\right)\right)}}, y - x, x\right) \]

      12. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{t - z}}{\mathsf{neg}\left(\left(a - t\right)\right)}, y - x, x\right) \]

      13. lift--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{t - z}{\mathsf{neg}\left(\color{blue}{\left(a - t\right)}\right)}, y - x, x\right) \]

      14. sub-negate-rev N/A

          \[\leadsto \mathsf{fma}\left(\frac{t - z}{\color{blue}{t - a}}, y - x, x\right) \]

      15. lower--.f64 83.5

          \[\leadsto \mathsf{fma}\left(\frac{t - z}{\color{blue}{t - a}}, y - x, x\right) \]

   3. Applied rewrites 83.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t - z}{t - a}, y - x, x\right)} \]

   4. Taylor expanded in t around 0

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{z}{a}}, y - x, x\right) \]
   5. Step-by-step derivation

      1. lower-/.f64 48.6

         \[\leadsto \mathsf{fma}\left(\frac{z}{\color{blue}{a}}, y - x, x\right) \]

   6. Applied rewrites 48.6%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{z}{a}}, y - x, x\right) \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 4: 70.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{z}{a}, y - x, x\right)\\ \mathbf{if}\;a \leq -1.18 \cdot 10^{-82}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 6000000:\\ \;\;\;\;\mathsf{fma}\left(y - x, \frac{a - z}{t}, y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ z a) (- y x) x)))
   (if (<= a -1.18e-82)
     t_1
     (if (<= a 6000000.0) (fma (- y x) (/ (- a z) t) y) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((z / a), (y - x), x);
	double tmp;
	if (a <= -1.18e-82) {
		tmp = t_1;
	} else if (a <= 6000000.0) {
		tmp = fma((y - x), ((a - z) / t), y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(z / a), Float64(y - x), x)
	tmp = 0.0
	if (a <= -1.18e-82)
		tmp = t_1;
	elseif (a <= 6000000.0)
		tmp = fma(Float64(y - x), Float64(Float64(a - z) / t), y);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z / a), $MachinePrecision] * N[(y - x), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[a, -1.18e-82], t$95$1, If[LessEqual[a, 6000000.0], N[(N[(y - x), $MachinePrecision] * N[(N[(a - z), $MachinePrecision] / t), $MachinePrecision] + y), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -1.1799999999999999e-82 or 6e6 < a

   1. Initial program 67.7%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} + x} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}} + x \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(y - x\right) \cdot \left(z - t\right)}}{a - t} + x \]

      5. associate-/l* N/A

         \[\leadsto \color{blue}{\left(y - x\right) \cdot \frac{z - t}{a - t}} + x \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\frac{z - t}{a - t} \cdot \left(y - x\right)} + x \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z - t}{a - t}, y - x, x\right)} \]

      8. frac-2neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\mathsf{neg}\left(\left(z - t\right)\right)}{\mathsf{neg}\left(\left(a - t\right)\right)}}, y - x, x\right) \]

      9. lift--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\mathsf{neg}\left(\color{blue}{\left(z - t\right)}\right)}{\mathsf{neg}\left(\left(a - t\right)\right)}, y - x, x\right) \]

      10. sub-negate-rev N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{t - z}}{\mathsf{neg}\left(\left(a - t\right)\right)}, y - x, x\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{t - z}{\mathsf{neg}\left(\left(a - t\right)\right)}}, y - x, x\right) \]

      12. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{t - z}}{\mathsf{neg}\left(\left(a - t\right)\right)}, y - x, x\right) \]

      13. lift--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{t - z}{\mathsf{neg}\left(\color{blue}{\left(a - t\right)}\right)}, y - x, x\right) \]

      14. sub-negate-rev N/A

          \[\leadsto \mathsf{fma}\left(\frac{t - z}{\color{blue}{t - a}}, y - x, x\right) \]

      15. lower--.f64 83.5

          \[\leadsto \mathsf{fma}\left(\frac{t - z}{\color{blue}{t - a}}, y - x, x\right) \]

   3. Applied rewrites 83.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t - z}{t - a}, y - x, x\right)} \]

   4. Taylor expanded in t around 0

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{z}{a}}, y - x, x\right) \]
   5. Step-by-step derivation

      1. lower-/.f64 48.6

         \[\leadsto \mathsf{fma}\left(\frac{z}{\color{blue}{a}}, y - x, x\right) \]

   6. Applied rewrites 48.6%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{z}{a}}, y - x, x\right) \]

   if -1.1799999999999999e-82 < a < 6e6

   1. Initial program 67.7%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]

   2. Taylor expanded in t around -inf

      \[\leadsto \color{blue}{y + -1 \cdot \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}} \]
   3. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto y + \color{blue}{-1 \cdot \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}} \]

      2. lower-*.f64 N/A

         \[\leadsto y + -1 \cdot \color{blue}{\frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}} \]

      3. lower-/.f64 N/A

         \[\leadsto y + -1 \cdot \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{\color{blue}{t}} \]

      4. lower--.f64 N/A

         \[\leadsto y + -1 \cdot \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t} \]

      5. lower-*.f64 N/A

         \[\leadsto y + -1 \cdot \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t} \]

      6. lower--.f64 N/A

         \[\leadsto y + -1 \cdot \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t} \]

      7. lower-*.f64 N/A

         \[\leadsto y + -1 \cdot \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t} \]

      8. lower--.f64 46.0

         \[\leadsto y + -1 \cdot \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t} \]

   4. Applied rewrites 46.0%

      \[\leadsto \color{blue}{y + -1 \cdot \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto y + \color{blue}{-1 \cdot \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}} \]

      2. +-commutative N/A

         \[\leadsto -1 \cdot \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t} + \color{blue}{y} \]

      3. add-flip N/A

         \[\leadsto -1 \cdot \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t} - \color{blue}{\left(\mathsf{neg}\left(y\right)\right)} \]

      4. sub-flip N/A

         \[\leadsto -1 \cdot \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t} + \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(y\right)\right)\right)\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto -1 \cdot \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t} + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}\right)\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right)\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{y}\right)\right)\right)\right) \]

      8. distribute-neg-frac N/A

         \[\leadsto \frac{\mathsf{neg}\left(\left(z \cdot \left(y - x\right) - a \cdot \left(y - x\right)\right)\right)}{t} + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right)\right) \]

      9. lift--.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\left(z \cdot \left(y - x\right) - a \cdot \left(y - x\right)\right)\right)}{t} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(y\right)\right)\right)\right) \]

      10. sub-negate-rev N/A

          \[\leadsto \frac{a \cdot \left(y - x\right) - z \cdot \left(y - x\right)}{t} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{y}\right)\right)\right)\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{a \cdot \left(y - x\right) - z \cdot \left(y - x\right)}{t} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(y\right)\right)\right)\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{a \cdot \left(y - x\right) - z \cdot \left(y - x\right)}{t} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(y\right)\right)\right)\right) \]

      13. distribute-rgt-out-- N/A

          \[\leadsto \frac{\left(y - x\right) \cdot \left(a - z\right)}{t} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{y}\right)\right)\right)\right) \]

      14. associate-/l* N/A

          \[\leadsto \left(y - x\right) \cdot \frac{a - z}{t} + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right)\right) \]

      15. remove-double-neg N/A

          \[\leadsto \left(y - x\right) \cdot \frac{a - z}{t} + y \]

      16. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(y - x, \color{blue}{\frac{a - z}{t}}, y\right) \]

   6. Applied rewrites 53.2%

      \[\leadsto \mathsf{fma}\left(y - x, \color{blue}{\frac{a - z}{t}}, y\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 66.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{a - t} \cdot y\\ \mathbf{if}\;t \leq -3.3 \cdot 10^{+19}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 3.75 \cdot 10^{+73}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, y - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (/ (- z t) (- a t)) y)))
   (if (<= t -3.3e+19) t_1 (if (<= t 3.75e+73) (fma (/ z a) (- y x) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = ((z - t) / (a - t)) * y;
	double tmp;
	if (t <= -3.3e+19) {
		tmp = t_1;
	} else if (t <= 3.75e+73) {
		tmp = fma((z / a), (y - x), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(z - t) / Float64(a - t)) * y)
	tmp = 0.0
	if (t <= -3.3e+19)
		tmp = t_1;
	elseif (t <= 3.75e+73)
		tmp = fma(Float64(z / a), Float64(y - x), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[t, -3.3e+19], t$95$1, If[LessEqual[t, 3.75e+73], N[(N[(z / a), $MachinePrecision] * N[(y - x), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -3.3e19 or 3.75e73 < t

   1. Initial program 67.7%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{y \cdot \left(z - t\right)}{a - t}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{y \cdot \left(z - t\right)}{\color{blue}{a - t}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{y \cdot \left(z - t\right)}{\color{blue}{a} - t} \]

      3. lower--.f64 N/A

         \[\leadsto \frac{y \cdot \left(z - t\right)}{a - t} \]

      4. lower--.f64 39.8

         \[\leadsto \frac{y \cdot \left(z - t\right)}{a - \color{blue}{t}} \]

   4. Applied rewrites 39.8%

      \[\leadsto \color{blue}{\frac{y \cdot \left(z - t\right)}{a - t}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{y \cdot \left(z - t\right)}{\color{blue}{a - t}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{y \cdot \left(z - t\right)}{\color{blue}{a} - t} \]

      3. associate-/l* N/A

         \[\leadsto y \cdot \color{blue}{\frac{z - t}{a - t}} \]

      4. lift--.f64 N/A

         \[\leadsto y \cdot \frac{z - t}{\color{blue}{a} - t} \]

      5. sub-negate-rev N/A

         \[\leadsto y \cdot \frac{\mathsf{neg}\left(\left(t - z\right)\right)}{\color{blue}{a} - t} \]

      6. lift--.f64 N/A

         \[\leadsto y \cdot \frac{\mathsf{neg}\left(\left(t - z\right)\right)}{a - t} \]

      7. lift--.f64 N/A

         \[\leadsto y \cdot \frac{\mathsf{neg}\left(\left(t - z\right)\right)}{a - \color{blue}{t}} \]

      8. sub-negate-rev N/A

         \[\leadsto y \cdot \frac{\mathsf{neg}\left(\left(t - z\right)\right)}{\mathsf{neg}\left(\left(t - a\right)\right)} \]

      9. lift--.f64 N/A

         \[\leadsto y \cdot \frac{\mathsf{neg}\left(\left(t - z\right)\right)}{\mathsf{neg}\left(\left(t - a\right)\right)} \]

      10. frac-2neg N/A

          \[\leadsto y \cdot \frac{t - z}{\color{blue}{t - a}} \]

      11. lift-/.f64 N/A

          \[\leadsto y \cdot \frac{t - z}{\color{blue}{t - a}} \]

      12. *-commutative N/A

          \[\leadsto \frac{t - z}{t - a} \cdot \color{blue}{y} \]

      13. lower-*.f64 52.1

          \[\leadsto \frac{t - z}{t - a} \cdot \color{blue}{y} \]

      14. lift-/.f64 N/A

          \[\leadsto \frac{t - z}{t - a} \cdot y \]

      15. frac-2neg N/A

          \[\leadsto \frac{\mathsf{neg}\left(\left(t - z\right)\right)}{\mathsf{neg}\left(\left(t - a\right)\right)} \cdot y \]

      16. lift--.f64 N/A

          \[\leadsto \frac{\mathsf{neg}\left(\left(t - z\right)\right)}{\mathsf{neg}\left(\left(t - a\right)\right)} \cdot y \]

      17. sub-negate-rev N/A

          \[\leadsto \frac{z - t}{\mathsf{neg}\left(\left(t - a\right)\right)} \cdot y \]

      18. lift--.f64 N/A

          \[\leadsto \frac{z - t}{\mathsf{neg}\left(\left(t - a\right)\right)} \cdot y \]

      19. lift--.f64 N/A

          \[\leadsto \frac{z - t}{\mathsf{neg}\left(\left(t - a\right)\right)} \cdot y \]

      20. sub-negate-rev N/A

          \[\leadsto \frac{z - t}{a - t} \cdot y \]

      21. lift--.f64 N/A

          \[\leadsto \frac{z - t}{a - t} \cdot y \]

      22. lower-/.f64 52.1

          \[\leadsto \frac{z - t}{a - t} \cdot y \]

   6. Applied rewrites 52.1%

      \[\leadsto \frac{z - t}{a - t} \cdot \color{blue}{y} \]

   if -3.3e19 < t < 3.75e73

   1. Initial program 67.7%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} + x} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}} + x \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(y - x\right) \cdot \left(z - t\right)}}{a - t} + x \]

      5. associate-/l* N/A

         \[\leadsto \color{blue}{\left(y - x\right) \cdot \frac{z - t}{a - t}} + x \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\frac{z - t}{a - t} \cdot \left(y - x\right)} + x \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z - t}{a - t}, y - x, x\right)} \]

      8. frac-2neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\mathsf{neg}\left(\left(z - t\right)\right)}{\mathsf{neg}\left(\left(a - t\right)\right)}}, y - x, x\right) \]

      9. lift--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\mathsf{neg}\left(\color{blue}{\left(z - t\right)}\right)}{\mathsf{neg}\left(\left(a - t\right)\right)}, y - x, x\right) \]

      10. sub-negate-rev N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{t - z}}{\mathsf{neg}\left(\left(a - t\right)\right)}, y - x, x\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{t - z}{\mathsf{neg}\left(\left(a - t\right)\right)}}, y - x, x\right) \]

      12. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{t - z}}{\mathsf{neg}\left(\left(a - t\right)\right)}, y - x, x\right) \]

      13. lift--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{t - z}{\mathsf{neg}\left(\color{blue}{\left(a - t\right)}\right)}, y - x, x\right) \]

      14. sub-negate-rev N/A

          \[\leadsto \mathsf{fma}\left(\frac{t - z}{\color{blue}{t - a}}, y - x, x\right) \]

      15. lower--.f64 83.5

          \[\leadsto \mathsf{fma}\left(\frac{t - z}{\color{blue}{t - a}}, y - x, x\right) \]

   3. Applied rewrites 83.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t - z}{t - a}, y - x, x\right)} \]

   4. Taylor expanded in t around 0

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{z}{a}}, y - x, x\right) \]
   5. Step-by-step derivation

      1. lower-/.f64 48.6

         \[\leadsto \mathsf{fma}\left(\frac{z}{\color{blue}{a}}, y - x, x\right) \]

   6. Applied rewrites 48.6%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{z}{a}}, y - x, x\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 63.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, \frac{a - z}{t}, y\right)\\ \mathbf{if}\;t \leq -3.9 \cdot 10^{+76}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 1.15 \cdot 10^{+83}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, y - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma y (/ (- a z) t) y)))
   (if (<= t -3.9e+76) t_1 (if (<= t 1.15e+83) (fma (/ z a) (- y x) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(y, ((a - z) / t), y);
	double tmp;
	if (t <= -3.9e+76) {
		tmp = t_1;
	} else if (t <= 1.15e+83) {
		tmp = fma((z / a), (y - x), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(y, Float64(Float64(a - z) / t), y)
	tmp = 0.0
	if (t <= -3.9e+76)
		tmp = t_1;
	elseif (t <= 1.15e+83)
		tmp = fma(Float64(z / a), Float64(y - x), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(y * N[(N[(a - z), $MachinePrecision] / t), $MachinePrecision] + y), $MachinePrecision]}, If[LessEqual[t, -3.9e+76], t$95$1, If[LessEqual[t, 1.15e+83], N[(N[(z / a), $MachinePrecision] * N[(y - x), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -3.89999999999999989e76 or 1.14999999999999997e83 < t

   1. Initial program 67.7%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]

   2. Taylor expanded in t around -inf

      \[\leadsto \color{blue}{y + -1 \cdot \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}} \]
   3. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto y + \color{blue}{-1 \cdot \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}} \]

      2. lower-*.f64 N/A

         \[\leadsto y + -1 \cdot \color{blue}{\frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}} \]

      3. lower-/.f64 N/A

         \[\leadsto y + -1 \cdot \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{\color{blue}{t}} \]

      4. lower--.f64 N/A

         \[\leadsto y + -1 \cdot \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t} \]

      5. lower-*.f64 N/A

         \[\leadsto y + -1 \cdot \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t} \]

      6. lower--.f64 N/A

         \[\leadsto y + -1 \cdot \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t} \]

      7. lower-*.f64 N/A

         \[\leadsto y + -1 \cdot \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t} \]

      8. lower--.f64 46.0

         \[\leadsto y + -1 \cdot \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t} \]

   4. Applied rewrites 46.0%

      \[\leadsto \color{blue}{y + -1 \cdot \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto y + \color{blue}{-1 \cdot \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}} \]

      2. +-commutative N/A

         \[\leadsto -1 \cdot \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t} + \color{blue}{y} \]

      3. add-flip N/A

         \[\leadsto -1 \cdot \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t} - \color{blue}{\left(\mathsf{neg}\left(y\right)\right)} \]

      4. sub-flip N/A

         \[\leadsto -1 \cdot \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t} + \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(y\right)\right)\right)\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto -1 \cdot \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t} + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}\right)\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right)\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{y}\right)\right)\right)\right) \]

      8. distribute-neg-frac N/A

         \[\leadsto \frac{\mathsf{neg}\left(\left(z \cdot \left(y - x\right) - a \cdot \left(y - x\right)\right)\right)}{t} + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right)\right) \]

      9. lift--.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\left(z \cdot \left(y - x\right) - a \cdot \left(y - x\right)\right)\right)}{t} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(y\right)\right)\right)\right) \]

      10. sub-negate-rev N/A

          \[\leadsto \frac{a \cdot \left(y - x\right) - z \cdot \left(y - x\right)}{t} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{y}\right)\right)\right)\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{a \cdot \left(y - x\right) - z \cdot \left(y - x\right)}{t} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(y\right)\right)\right)\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{a \cdot \left(y - x\right) - z \cdot \left(y - x\right)}{t} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(y\right)\right)\right)\right) \]

      13. distribute-rgt-out-- N/A

          \[\leadsto \frac{\left(y - x\right) \cdot \left(a - z\right)}{t} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{y}\right)\right)\right)\right) \]

      14. associate-/l* N/A

          \[\leadsto \left(y - x\right) \cdot \frac{a - z}{t} + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right)\right) \]

      15. remove-double-neg N/A

          \[\leadsto \left(y - x\right) \cdot \frac{a - z}{t} + y \]

      16. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(y - x, \color{blue}{\frac{a - z}{t}}, y\right) \]

   6. Applied rewrites 53.2%

      \[\leadsto \mathsf{fma}\left(y - x, \color{blue}{\frac{a - z}{t}}, y\right) \]

   7. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(y, \frac{\color{blue}{a - z}}{t}, y\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 36.2%

      \[\leadsto \mathsf{fma}\left(y, \frac{\color{blue}{a - z}}{t}, y\right) \]

   if -3.89999999999999989e76 < t < 1.14999999999999997e83

   1. Initial program 67.7%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} + x} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}} + x \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(y - x\right) \cdot \left(z - t\right)}}{a - t} + x \]

      5. associate-/l* N/A

         \[\leadsto \color{blue}{\left(y - x\right) \cdot \frac{z - t}{a - t}} + x \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\frac{z - t}{a - t} \cdot \left(y - x\right)} + x \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z - t}{a - t}, y - x, x\right)} \]

      8. frac-2neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\mathsf{neg}\left(\left(z - t\right)\right)}{\mathsf{neg}\left(\left(a - t\right)\right)}}, y - x, x\right) \]

      9. lift--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\mathsf{neg}\left(\color{blue}{\left(z - t\right)}\right)}{\mathsf{neg}\left(\left(a - t\right)\right)}, y - x, x\right) \]

      10. sub-negate-rev N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{t - z}}{\mathsf{neg}\left(\left(a - t\right)\right)}, y - x, x\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{t - z}{\mathsf{neg}\left(\left(a - t\right)\right)}}, y - x, x\right) \]

      12. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{t - z}}{\mathsf{neg}\left(\left(a - t\right)\right)}, y - x, x\right) \]

      13. lift--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{t - z}{\mathsf{neg}\left(\color{blue}{\left(a - t\right)}\right)}, y - x, x\right) \]

      14. sub-negate-rev N/A

          \[\leadsto \mathsf{fma}\left(\frac{t - z}{\color{blue}{t - a}}, y - x, x\right) \]

      15. lower--.f64 83.5

          \[\leadsto \mathsf{fma}\left(\frac{t - z}{\color{blue}{t - a}}, y - x, x\right) \]

   3. Applied rewrites 83.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t - z}{t - a}, y - x, x\right)} \]

   4. Taylor expanded in t around 0

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{z}{a}}, y - x, x\right) \]
   5. Step-by-step derivation

      1. lower-/.f64 48.6

         \[\leadsto \mathsf{fma}\left(\frac{z}{\color{blue}{a}}, y - x, x\right) \]

   6. Applied rewrites 48.6%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{z}{a}}, y - x, x\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 56.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, \frac{a - z}{t}, y\right)\\ \mathbf{if}\;t \leq -3.9 \cdot 10^{+76}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 1.15 \cdot 10^{+83}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma y (/ (- a z) t) y)))
   (if (<= t -3.9e+76) t_1 (if (<= t 1.15e+83) (fma (/ z a) y x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(y, ((a - z) / t), y);
	double tmp;
	if (t <= -3.9e+76) {
		tmp = t_1;
	} else if (t <= 1.15e+83) {
		tmp = fma((z / a), y, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(y, Float64(Float64(a - z) / t), y)
	tmp = 0.0
	if (t <= -3.9e+76)
		tmp = t_1;
	elseif (t <= 1.15e+83)
		tmp = fma(Float64(z / a), y, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(y * N[(N[(a - z), $MachinePrecision] / t), $MachinePrecision] + y), $MachinePrecision]}, If[LessEqual[t, -3.9e+76], t$95$1, If[LessEqual[t, 1.15e+83], N[(N[(z / a), $MachinePrecision] * y + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -3.89999999999999989e76 or 1.14999999999999997e83 < t

   1. Initial program 67.7%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]

   2. Taylor expanded in t around -inf

      \[\leadsto \color{blue}{y + -1 \cdot \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}} \]
   3. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto y + \color{blue}{-1 \cdot \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}} \]

      2. lower-*.f64 N/A

         \[\leadsto y + -1 \cdot \color{blue}{\frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}} \]

      3. lower-/.f64 N/A

         \[\leadsto y + -1 \cdot \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{\color{blue}{t}} \]

      4. lower--.f64 N/A

         \[\leadsto y + -1 \cdot \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t} \]

      5. lower-*.f64 N/A

         \[\leadsto y + -1 \cdot \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t} \]

      6. lower--.f64 N/A

         \[\leadsto y + -1 \cdot \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t} \]

      7. lower-*.f64 N/A

         \[\leadsto y + -1 \cdot \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t} \]

      8. lower--.f64 46.0

         \[\leadsto y + -1 \cdot \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t} \]

   4. Applied rewrites 46.0%

      \[\leadsto \color{blue}{y + -1 \cdot \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto y + \color{blue}{-1 \cdot \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}} \]

      2. +-commutative N/A

         \[\leadsto -1 \cdot \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t} + \color{blue}{y} \]

      3. add-flip N/A

         \[\leadsto -1 \cdot \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t} - \color{blue}{\left(\mathsf{neg}\left(y\right)\right)} \]

      4. sub-flip N/A

         \[\leadsto -1 \cdot \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t} + \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(y\right)\right)\right)\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto -1 \cdot \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t} + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}\right)\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right)\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{y}\right)\right)\right)\right) \]

      8. distribute-neg-frac N/A

         \[\leadsto \frac{\mathsf{neg}\left(\left(z \cdot \left(y - x\right) - a \cdot \left(y - x\right)\right)\right)}{t} + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right)\right) \]

      9. lift--.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\left(z \cdot \left(y - x\right) - a \cdot \left(y - x\right)\right)\right)}{t} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(y\right)\right)\right)\right) \]

      10. sub-negate-rev N/A

          \[\leadsto \frac{a \cdot \left(y - x\right) - z \cdot \left(y - x\right)}{t} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{y}\right)\right)\right)\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{a \cdot \left(y - x\right) - z \cdot \left(y - x\right)}{t} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(y\right)\right)\right)\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{a \cdot \left(y - x\right) - z \cdot \left(y - x\right)}{t} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(y\right)\right)\right)\right) \]

      13. distribute-rgt-out-- N/A

          \[\leadsto \frac{\left(y - x\right) \cdot \left(a - z\right)}{t} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{y}\right)\right)\right)\right) \]

      14. associate-/l* N/A

          \[\leadsto \left(y - x\right) \cdot \frac{a - z}{t} + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right)\right) \]

      15. remove-double-neg N/A

          \[\leadsto \left(y - x\right) \cdot \frac{a - z}{t} + y \]

      16. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(y - x, \color{blue}{\frac{a - z}{t}}, y\right) \]

   6. Applied rewrites 53.2%

      \[\leadsto \mathsf{fma}\left(y - x, \color{blue}{\frac{a - z}{t}}, y\right) \]

   7. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(y, \frac{\color{blue}{a - z}}{t}, y\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 36.2%

      \[\leadsto \mathsf{fma}\left(y, \frac{\color{blue}{a - z}}{t}, y\right) \]

   if -3.89999999999999989e76 < t < 1.14999999999999997e83

   1. Initial program 67.7%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} + x} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}} + x \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(y - x\right) \cdot \left(z - t\right)}}{a - t} + x \]

      5. associate-/l* N/A

         \[\leadsto \color{blue}{\left(y - x\right) \cdot \frac{z - t}{a - t}} + x \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\frac{z - t}{a - t} \cdot \left(y - x\right)} + x \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z - t}{a - t}, y - x, x\right)} \]

      8. frac-2neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\mathsf{neg}\left(\left(z - t\right)\right)}{\mathsf{neg}\left(\left(a - t\right)\right)}}, y - x, x\right) \]

      9. lift--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\mathsf{neg}\left(\color{blue}{\left(z - t\right)}\right)}{\mathsf{neg}\left(\left(a - t\right)\right)}, y - x, x\right) \]

      10. sub-negate-rev N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{t - z}}{\mathsf{neg}\left(\left(a - t\right)\right)}, y - x, x\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{t - z}{\mathsf{neg}\left(\left(a - t\right)\right)}}, y - x, x\right) \]

      12. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{t - z}}{\mathsf{neg}\left(\left(a - t\right)\right)}, y - x, x\right) \]

      13. lift--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{t - z}{\mathsf{neg}\left(\color{blue}{\left(a - t\right)}\right)}, y - x, x\right) \]

      14. sub-negate-rev N/A

          \[\leadsto \mathsf{fma}\left(\frac{t - z}{\color{blue}{t - a}}, y - x, x\right) \]

      15. lower--.f64 83.5

          \[\leadsto \mathsf{fma}\left(\frac{t - z}{\color{blue}{t - a}}, y - x, x\right) \]

   3. Applied rewrites 83.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t - z}{t - a}, y - x, x\right)} \]

   4. Taylor expanded in t around 0

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{z}{a}}, y - x, x\right) \]
   5. Step-by-step derivation

      1. lower-/.f64 48.6

         \[\leadsto \mathsf{fma}\left(\frac{z}{\color{blue}{a}}, y - x, x\right) \]

   6. Applied rewrites 48.6%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{z}{a}}, y - x, x\right) \]

   7. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(\frac{z}{a}, \color{blue}{y}, x\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 41.3%

      \[\leadsto \mathsf{fma}\left(\frac{z}{a}, \color{blue}{y}, x\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 53.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -9.5 \cdot 10^{+120}:\\ \;\;\;\;1 \cdot y\\ \mathbf{elif}\;t \leq 1.16 \cdot 10^{+83}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -9.5e+120)
   (* 1.0 y)
   (if (<= t 1.16e+83) (fma (/ z a) y x) (* 1.0 y))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -9.5e+120) {
		tmp = 1.0 * y;
	} else if (t <= 1.16e+83) {
		tmp = fma((z / a), y, x);
	} else {
		tmp = 1.0 * y;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -9.5e+120)
		tmp = Float64(1.0 * y);
	elseif (t <= 1.16e+83)
		tmp = fma(Float64(z / a), y, x);
	else
		tmp = Float64(1.0 * y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -9.5e+120], N[(1.0 * y), $MachinePrecision], If[LessEqual[t, 1.16e+83], N[(N[(z / a), $MachinePrecision] * y + x), $MachinePrecision], N[(1.0 * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < -9.5e120 or 1.1600000000000001e83 < t

   1. Initial program 67.7%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{y \cdot \left(z - t\right)}{a - t}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{y \cdot \left(z - t\right)}{\color{blue}{a - t}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{y \cdot \left(z - t\right)}{\color{blue}{a} - t} \]

      3. lower--.f64 N/A

         \[\leadsto \frac{y \cdot \left(z - t\right)}{a - t} \]

      4. lower--.f64 39.8

         \[\leadsto \frac{y \cdot \left(z - t\right)}{a - \color{blue}{t}} \]

   4. Applied rewrites 39.8%

      \[\leadsto \color{blue}{\frac{y \cdot \left(z - t\right)}{a - t}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{y \cdot \left(z - t\right)}{\color{blue}{a - t}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{y \cdot \left(z - t\right)}{\color{blue}{a} - t} \]

      3. associate-/l* N/A

         \[\leadsto y \cdot \color{blue}{\frac{z - t}{a - t}} \]

      4. lift--.f64 N/A

         \[\leadsto y \cdot \frac{z - t}{\color{blue}{a} - t} \]

      5. sub-negate-rev N/A

         \[\leadsto y \cdot \frac{\mathsf{neg}\left(\left(t - z\right)\right)}{\color{blue}{a} - t} \]

      6. lift--.f64 N/A

         \[\leadsto y \cdot \frac{\mathsf{neg}\left(\left(t - z\right)\right)}{a - t} \]

      7. lift--.f64 N/A

         \[\leadsto y \cdot \frac{\mathsf{neg}\left(\left(t - z\right)\right)}{a - \color{blue}{t}} \]

      8. sub-negate-rev N/A

         \[\leadsto y \cdot \frac{\mathsf{neg}\left(\left(t - z\right)\right)}{\mathsf{neg}\left(\left(t - a\right)\right)} \]

      9. lift--.f64 N/A

         \[\leadsto y \cdot \frac{\mathsf{neg}\left(\left(t - z\right)\right)}{\mathsf{neg}\left(\left(t - a\right)\right)} \]

      10. frac-2neg N/A

          \[\leadsto y \cdot \frac{t - z}{\color{blue}{t - a}} \]

      11. lift-/.f64 N/A

          \[\leadsto y \cdot \frac{t - z}{\color{blue}{t - a}} \]

      12. *-commutative N/A

          \[\leadsto \frac{t - z}{t - a} \cdot \color{blue}{y} \]

      13. lower-*.f64 52.1

          \[\leadsto \frac{t - z}{t - a} \cdot \color{blue}{y} \]

      14. lift-/.f64 N/A

          \[\leadsto \frac{t - z}{t - a} \cdot y \]

      15. frac-2neg N/A

          \[\leadsto \frac{\mathsf{neg}\left(\left(t - z\right)\right)}{\mathsf{neg}\left(\left(t - a\right)\right)} \cdot y \]

      16. lift--.f64 N/A

          \[\leadsto \frac{\mathsf{neg}\left(\left(t - z\right)\right)}{\mathsf{neg}\left(\left(t - a\right)\right)} \cdot y \]

      17. sub-negate-rev N/A

          \[\leadsto \frac{z - t}{\mathsf{neg}\left(\left(t - a\right)\right)} \cdot y \]

      18. lift--.f64 N/A

          \[\leadsto \frac{z - t}{\mathsf{neg}\left(\left(t - a\right)\right)} \cdot y \]

      19. lift--.f64 N/A

          \[\leadsto \frac{z - t}{\mathsf{neg}\left(\left(t - a\right)\right)} \cdot y \]

      20. sub-negate-rev N/A

          \[\leadsto \frac{z - t}{a - t} \cdot y \]

      21. lift--.f64 N/A

          \[\leadsto \frac{z - t}{a - t} \cdot y \]

      22. lower-/.f64 52.1

          \[\leadsto \frac{z - t}{a - t} \cdot y \]

   6. Applied rewrites 52.1%

      \[\leadsto \frac{z - t}{a - t} \cdot \color{blue}{y} \]

   7. Taylor expanded in t around inf

      \[\leadsto 1 \cdot y \]
   8. Step-by-step derivation

   9. Applied rewrites 25.4%

      \[\leadsto 1 \cdot y \]

   if -9.5e120 < t < 1.1600000000000001e83

   1. Initial program 67.7%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} + x} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}} + x \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(y - x\right) \cdot \left(z - t\right)}}{a - t} + x \]

      5. associate-/l* N/A

         \[\leadsto \color{blue}{\left(y - x\right) \cdot \frac{z - t}{a - t}} + x \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\frac{z - t}{a - t} \cdot \left(y - x\right)} + x \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z - t}{a - t}, y - x, x\right)} \]

      8. frac-2neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\mathsf{neg}\left(\left(z - t\right)\right)}{\mathsf{neg}\left(\left(a - t\right)\right)}}, y - x, x\right) \]

      9. lift--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\mathsf{neg}\left(\color{blue}{\left(z - t\right)}\right)}{\mathsf{neg}\left(\left(a - t\right)\right)}, y - x, x\right) \]

      10. sub-negate-rev N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{t - z}}{\mathsf{neg}\left(\left(a - t\right)\right)}, y - x, x\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{t - z}{\mathsf{neg}\left(\left(a - t\right)\right)}}, y - x, x\right) \]

      12. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{t - z}}{\mathsf{neg}\left(\left(a - t\right)\right)}, y - x, x\right) \]

      13. lift--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{t - z}{\mathsf{neg}\left(\color{blue}{\left(a - t\right)}\right)}, y - x, x\right) \]

      14. sub-negate-rev N/A

          \[\leadsto \mathsf{fma}\left(\frac{t - z}{\color{blue}{t - a}}, y - x, x\right) \]

      15. lower--.f64 83.5

          \[\leadsto \mathsf{fma}\left(\frac{t - z}{\color{blue}{t - a}}, y - x, x\right) \]

   3. Applied rewrites 83.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t - z}{t - a}, y - x, x\right)} \]

   4. Taylor expanded in t around 0

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{z}{a}}, y - x, x\right) \]
   5. Step-by-step derivation

      1. lower-/.f64 48.6

         \[\leadsto \mathsf{fma}\left(\frac{z}{\color{blue}{a}}, y - x, x\right) \]

   6. Applied rewrites 48.6%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{z}{a}}, y - x, x\right) \]

   7. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(\frac{z}{a}, \color{blue}{y}, x\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 41.3%

      \[\leadsto \mathsf{fma}\left(\frac{z}{a}, \color{blue}{y}, x\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 36.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -3.1 \cdot 10^{+17}:\\ \;\;\;\;1 \cdot y\\ \mathbf{elif}\;t \leq 1.25 \cdot 10^{+17}:\\ \;\;\;\;\frac{z}{a} \cdot y\\ \mathbf{else}:\\ \;\;\;\;1 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -3.1e+17) (* 1.0 y) (if (<= t 1.25e+17) (* (/ z a) y) (* 1.0 y))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -3.1e+17) {
		tmp = 1.0 * y;
	} else if (t <= 1.25e+17) {
		tmp = (z / a) * y;
	} else {
		tmp = 1.0 * y;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (t <= (-3.1d+17)) then
        tmp = 1.0d0 * y
    else if (t <= 1.25d+17) then
        tmp = (z / a) * y
    else
        tmp = 1.0d0 * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -3.1e+17) {
		tmp = 1.0 * y;
	} else if (t <= 1.25e+17) {
		tmp = (z / a) * y;
	} else {
		tmp = 1.0 * y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= -3.1e+17:
		tmp = 1.0 * y
	elif t <= 1.25e+17:
		tmp = (z / a) * y
	else:
		tmp = 1.0 * y
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -3.1e+17)
		tmp = Float64(1.0 * y);
	elseif (t <= 1.25e+17)
		tmp = Float64(Float64(z / a) * y);
	else
		tmp = Float64(1.0 * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= -3.1e+17)
		tmp = 1.0 * y;
	elseif (t <= 1.25e+17)
		tmp = (z / a) * y;
	else
		tmp = 1.0 * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -3.1e+17], N[(1.0 * y), $MachinePrecision], If[LessEqual[t, 1.25e+17], N[(N[(z / a), $MachinePrecision] * y), $MachinePrecision], N[(1.0 * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < -3.1e17 or 1.25e17 < t

   1. Initial program 67.7%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{y \cdot \left(z - t\right)}{a - t}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{y \cdot \left(z - t\right)}{\color{blue}{a - t}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{y \cdot \left(z - t\right)}{\color{blue}{a} - t} \]

      3. lower--.f64 N/A

         \[\leadsto \frac{y \cdot \left(z - t\right)}{a - t} \]

      4. lower--.f64 39.8

         \[\leadsto \frac{y \cdot \left(z - t\right)}{a - \color{blue}{t}} \]

   4. Applied rewrites 39.8%

      \[\leadsto \color{blue}{\frac{y \cdot \left(z - t\right)}{a - t}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{y \cdot \left(z - t\right)}{\color{blue}{a - t}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{y \cdot \left(z - t\right)}{\color{blue}{a} - t} \]

      3. associate-/l* N/A

         \[\leadsto y \cdot \color{blue}{\frac{z - t}{a - t}} \]

      4. lift--.f64 N/A

         \[\leadsto y \cdot \frac{z - t}{\color{blue}{a} - t} \]

      5. sub-negate-rev N/A

         \[\leadsto y \cdot \frac{\mathsf{neg}\left(\left(t - z\right)\right)}{\color{blue}{a} - t} \]

      6. lift--.f64 N/A

         \[\leadsto y \cdot \frac{\mathsf{neg}\left(\left(t - z\right)\right)}{a - t} \]

      7. lift--.f64 N/A

         \[\leadsto y \cdot \frac{\mathsf{neg}\left(\left(t - z\right)\right)}{a - \color{blue}{t}} \]

      8. sub-negate-rev N/A

         \[\leadsto y \cdot \frac{\mathsf{neg}\left(\left(t - z\right)\right)}{\mathsf{neg}\left(\left(t - a\right)\right)} \]

      9. lift--.f64 N/A

         \[\leadsto y \cdot \frac{\mathsf{neg}\left(\left(t - z\right)\right)}{\mathsf{neg}\left(\left(t - a\right)\right)} \]

      10. frac-2neg N/A

          \[\leadsto y \cdot \frac{t - z}{\color{blue}{t - a}} \]

      11. lift-/.f64 N/A

          \[\leadsto y \cdot \frac{t - z}{\color{blue}{t - a}} \]

      12. *-commutative N/A

          \[\leadsto \frac{t - z}{t - a} \cdot \color{blue}{y} \]

      13. lower-*.f64 52.1

          \[\leadsto \frac{t - z}{t - a} \cdot \color{blue}{y} \]

      14. lift-/.f64 N/A

          \[\leadsto \frac{t - z}{t - a} \cdot y \]

      15. frac-2neg N/A

          \[\leadsto \frac{\mathsf{neg}\left(\left(t - z\right)\right)}{\mathsf{neg}\left(\left(t - a\right)\right)} \cdot y \]

      16. lift--.f64 N/A

          \[\leadsto \frac{\mathsf{neg}\left(\left(t - z\right)\right)}{\mathsf{neg}\left(\left(t - a\right)\right)} \cdot y \]

      17. sub-negate-rev N/A

          \[\leadsto \frac{z - t}{\mathsf{neg}\left(\left(t - a\right)\right)} \cdot y \]

      18. lift--.f64 N/A

          \[\leadsto \frac{z - t}{\mathsf{neg}\left(\left(t - a\right)\right)} \cdot y \]

      19. lift--.f64 N/A

          \[\leadsto \frac{z - t}{\mathsf{neg}\left(\left(t - a\right)\right)} \cdot y \]

      20. sub-negate-rev N/A

          \[\leadsto \frac{z - t}{a - t} \cdot y \]

      21. lift--.f64 N/A

          \[\leadsto \frac{z - t}{a - t} \cdot y \]

      22. lower-/.f64 52.1

          \[\leadsto \frac{z - t}{a - t} \cdot y \]

   6. Applied rewrites 52.1%

      \[\leadsto \frac{z - t}{a - t} \cdot \color{blue}{y} \]

   7. Taylor expanded in t around inf

      \[\leadsto 1 \cdot y \]
   8. Step-by-step derivation

   9. Applied rewrites 25.4%

      \[\leadsto 1 \cdot y \]

   if -3.1e17 < t < 1.25e17

   1. Initial program 67.7%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{y \cdot \left(z - t\right)}{a - t}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{y \cdot \left(z - t\right)}{\color{blue}{a - t}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{y \cdot \left(z - t\right)}{\color{blue}{a} - t} \]

      3. lower--.f64 N/A

         \[\leadsto \frac{y \cdot \left(z - t\right)}{a - t} \]

      4. lower--.f64 39.8

         \[\leadsto \frac{y \cdot \left(z - t\right)}{a - \color{blue}{t}} \]

   4. Applied rewrites 39.8%

      \[\leadsto \color{blue}{\frac{y \cdot \left(z - t\right)}{a - t}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{y \cdot \left(z - t\right)}{\color{blue}{a - t}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{y \cdot \left(z - t\right)}{\color{blue}{a} - t} \]

      3. associate-/l* N/A

         \[\leadsto y \cdot \color{blue}{\frac{z - t}{a - t}} \]

      4. lift--.f64 N/A

         \[\leadsto y \cdot \frac{z - t}{\color{blue}{a} - t} \]

      5. sub-negate-rev N/A

         \[\leadsto y \cdot \frac{\mathsf{neg}\left(\left(t - z\right)\right)}{\color{blue}{a} - t} \]

      6. lift--.f64 N/A

         \[\leadsto y \cdot \frac{\mathsf{neg}\left(\left(t - z\right)\right)}{a - t} \]

      7. lift--.f64 N/A

         \[\leadsto y \cdot \frac{\mathsf{neg}\left(\left(t - z\right)\right)}{a - \color{blue}{t}} \]

      8. sub-negate-rev N/A

         \[\leadsto y \cdot \frac{\mathsf{neg}\left(\left(t - z\right)\right)}{\mathsf{neg}\left(\left(t - a\right)\right)} \]

      9. lift--.f64 N/A

         \[\leadsto y \cdot \frac{\mathsf{neg}\left(\left(t - z\right)\right)}{\mathsf{neg}\left(\left(t - a\right)\right)} \]

      10. frac-2neg N/A

          \[\leadsto y \cdot \frac{t - z}{\color{blue}{t - a}} \]

      11. lift-/.f64 N/A

          \[\leadsto y \cdot \frac{t - z}{\color{blue}{t - a}} \]

      12. *-commutative N/A

          \[\leadsto \frac{t - z}{t - a} \cdot \color{blue}{y} \]

      13. lower-*.f64 52.1

          \[\leadsto \frac{t - z}{t - a} \cdot \color{blue}{y} \]

      14. lift-/.f64 N/A

          \[\leadsto \frac{t - z}{t - a} \cdot y \]

      15. frac-2neg N/A

          \[\leadsto \frac{\mathsf{neg}\left(\left(t - z\right)\right)}{\mathsf{neg}\left(\left(t - a\right)\right)} \cdot y \]

      16. lift--.f64 N/A

          \[\leadsto \frac{\mathsf{neg}\left(\left(t - z\right)\right)}{\mathsf{neg}\left(\left(t - a\right)\right)} \cdot y \]

      17. sub-negate-rev N/A

          \[\leadsto \frac{z - t}{\mathsf{neg}\left(\left(t - a\right)\right)} \cdot y \]

      18. lift--.f64 N/A

          \[\leadsto \frac{z - t}{\mathsf{neg}\left(\left(t - a\right)\right)} \cdot y \]

      19. lift--.f64 N/A

          \[\leadsto \frac{z - t}{\mathsf{neg}\left(\left(t - a\right)\right)} \cdot y \]

      20. sub-negate-rev N/A

          \[\leadsto \frac{z - t}{a - t} \cdot y \]

      21. lift--.f64 N/A

          \[\leadsto \frac{z - t}{a - t} \cdot y \]

      22. lower-/.f64 52.1

          \[\leadsto \frac{z - t}{a - t} \cdot y \]

   6. Applied rewrites 52.1%

      \[\leadsto \frac{z - t}{a - t} \cdot \color{blue}{y} \]

   7. Taylor expanded in t around 0

      \[\leadsto \frac{z}{a} \cdot y \]
   8. Step-by-step derivation

      1. lower-/.f64 19.5

         \[\leadsto \frac{z}{a} \cdot y \]

   9. Applied rewrites 19.5%

      \[\leadsto \frac{z}{a} \cdot y \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 35.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2.2 \cdot 10^{+17}:\\ \;\;\;\;1 \cdot y\\ \mathbf{elif}\;t \leq 1.25 \cdot 10^{+17}:\\ \;\;\;\;z \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -2.2e+17) (* 1.0 y) (if (<= t 1.25e+17) (* z (/ y a)) (* 1.0 y))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -2.2e+17) {
		tmp = 1.0 * y;
	} else if (t <= 1.25e+17) {
		tmp = z * (y / a);
	} else {
		tmp = 1.0 * y;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (t <= (-2.2d+17)) then
        tmp = 1.0d0 * y
    else if (t <= 1.25d+17) then
        tmp = z * (y / a)
    else
        tmp = 1.0d0 * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -2.2e+17) {
		tmp = 1.0 * y;
	} else if (t <= 1.25e+17) {
		tmp = z * (y / a);
	} else {
		tmp = 1.0 * y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= -2.2e+17:
		tmp = 1.0 * y
	elif t <= 1.25e+17:
		tmp = z * (y / a)
	else:
		tmp = 1.0 * y
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -2.2e+17)
		tmp = Float64(1.0 * y);
	elseif (t <= 1.25e+17)
		tmp = Float64(z * Float64(y / a));
	else
		tmp = Float64(1.0 * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= -2.2e+17)
		tmp = 1.0 * y;
	elseif (t <= 1.25e+17)
		tmp = z * (y / a);
	else
		tmp = 1.0 * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -2.2e+17], N[(1.0 * y), $MachinePrecision], If[LessEqual[t, 1.25e+17], N[(z * N[(y / a), $MachinePrecision]), $MachinePrecision], N[(1.0 * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < -2.2e17 or 1.25e17 < t

   1. Initial program 67.7%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{y \cdot \left(z - t\right)}{a - t}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{y \cdot \left(z - t\right)}{\color{blue}{a - t}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{y \cdot \left(z - t\right)}{\color{blue}{a} - t} \]

      3. lower--.f64 N/A

         \[\leadsto \frac{y \cdot \left(z - t\right)}{a - t} \]

      4. lower--.f64 39.8

         \[\leadsto \frac{y \cdot \left(z - t\right)}{a - \color{blue}{t}} \]

   4. Applied rewrites 39.8%

      \[\leadsto \color{blue}{\frac{y \cdot \left(z - t\right)}{a - t}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{y \cdot \left(z - t\right)}{\color{blue}{a - t}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{y \cdot \left(z - t\right)}{\color{blue}{a} - t} \]

      3. associate-/l* N/A

         \[\leadsto y \cdot \color{blue}{\frac{z - t}{a - t}} \]

      4. lift--.f64 N/A

         \[\leadsto y \cdot \frac{z - t}{\color{blue}{a} - t} \]

      5. sub-negate-rev N/A

         \[\leadsto y \cdot \frac{\mathsf{neg}\left(\left(t - z\right)\right)}{\color{blue}{a} - t} \]

      6. lift--.f64 N/A

         \[\leadsto y \cdot \frac{\mathsf{neg}\left(\left(t - z\right)\right)}{a - t} \]

      7. lift--.f64 N/A

         \[\leadsto y \cdot \frac{\mathsf{neg}\left(\left(t - z\right)\right)}{a - \color{blue}{t}} \]

      8. sub-negate-rev N/A

         \[\leadsto y \cdot \frac{\mathsf{neg}\left(\left(t - z\right)\right)}{\mathsf{neg}\left(\left(t - a\right)\right)} \]

      9. lift--.f64 N/A

         \[\leadsto y \cdot \frac{\mathsf{neg}\left(\left(t - z\right)\right)}{\mathsf{neg}\left(\left(t - a\right)\right)} \]

      10. frac-2neg N/A

          \[\leadsto y \cdot \frac{t - z}{\color{blue}{t - a}} \]

      11. lift-/.f64 N/A

          \[\leadsto y \cdot \frac{t - z}{\color{blue}{t - a}} \]

      12. *-commutative N/A

          \[\leadsto \frac{t - z}{t - a} \cdot \color{blue}{y} \]

      13. lower-*.f64 52.1

          \[\leadsto \frac{t - z}{t - a} \cdot \color{blue}{y} \]

      14. lift-/.f64 N/A

          \[\leadsto \frac{t - z}{t - a} \cdot y \]

      15. frac-2neg N/A

          \[\leadsto \frac{\mathsf{neg}\left(\left(t - z\right)\right)}{\mathsf{neg}\left(\left(t - a\right)\right)} \cdot y \]

      16. lift--.f64 N/A

          \[\leadsto \frac{\mathsf{neg}\left(\left(t - z\right)\right)}{\mathsf{neg}\left(\left(t - a\right)\right)} \cdot y \]

      17. sub-negate-rev N/A

          \[\leadsto \frac{z - t}{\mathsf{neg}\left(\left(t - a\right)\right)} \cdot y \]

      18. lift--.f64 N/A

          \[\leadsto \frac{z - t}{\mathsf{neg}\left(\left(t - a\right)\right)} \cdot y \]

      19. lift--.f64 N/A

          \[\leadsto \frac{z - t}{\mathsf{neg}\left(\left(t - a\right)\right)} \cdot y \]

      20. sub-negate-rev N/A

          \[\leadsto \frac{z - t}{a - t} \cdot y \]

      21. lift--.f64 N/A

          \[\leadsto \frac{z - t}{a - t} \cdot y \]

      22. lower-/.f64 52.1

          \[\leadsto \frac{z - t}{a - t} \cdot y \]

   6. Applied rewrites 52.1%

      \[\leadsto \frac{z - t}{a - t} \cdot \color{blue}{y} \]

   7. Taylor expanded in t around inf

      \[\leadsto 1 \cdot y \]
   8. Step-by-step derivation

   9. Applied rewrites 25.4%

      \[\leadsto 1 \cdot y \]

   if -2.2e17 < t < 1.25e17

   1. Initial program 67.7%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]

   2. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(\frac{y}{a - t} - \frac{x}{a - t}\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto z \cdot \color{blue}{\left(\frac{y}{a - t} - \frac{x}{a - t}\right)} \]

      2. lower--.f64 N/A

         \[\leadsto z \cdot \left(\frac{y}{a - t} - \color{blue}{\frac{x}{a - t}}\right) \]

      3. lower-/.f64 N/A

         \[\leadsto z \cdot \left(\frac{y}{a - t} - \frac{\color{blue}{x}}{a - t}\right) \]

      4. lower--.f64 N/A

         \[\leadsto z \cdot \left(\frac{y}{a - t} - \frac{x}{a - t}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto z \cdot \left(\frac{y}{a - t} - \frac{x}{\color{blue}{a - t}}\right) \]

      6. lower--.f64 41.7

         \[\leadsto z \cdot \left(\frac{y}{a - t} - \frac{x}{a - \color{blue}{t}}\right) \]

   4. Applied rewrites 41.7%

      \[\leadsto \color{blue}{z \cdot \left(\frac{y}{a - t} - \frac{x}{a - t}\right)} \]

   5. Taylor expanded in x around 0

      \[\leadsto z \cdot \frac{y}{\color{blue}{a - t}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto z \cdot \frac{y}{a - \color{blue}{t}} \]

      2. lower--.f64 23.7

         \[\leadsto z \cdot \frac{y}{a - t} \]

   7. Applied rewrites 23.7%

      \[\leadsto z \cdot \frac{y}{\color{blue}{a - t}} \]

   8. Taylor expanded in t around 0

      \[\leadsto z \cdot \frac{y}{a} \]
   9. Step-by-step derivation

   10. Applied rewrites 18.5%

       \[\leadsto z \cdot \frac{y}{a} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 33.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -3.1 \cdot 10^{+17}:\\ \;\;\;\;1 \cdot y\\ \mathbf{elif}\;t \leq 4 \cdot 10^{+69}:\\ \;\;\;\;\frac{y \cdot z}{a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -3.1e+17) (* 1.0 y) (if (<= t 4e+69) (/ (* y z) a) (* 1.0 y))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -3.1e+17) {
		tmp = 1.0 * y;
	} else if (t <= 4e+69) {
		tmp = (y * z) / a;
	} else {
		tmp = 1.0 * y;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (t <= (-3.1d+17)) then
        tmp = 1.0d0 * y
    else if (t <= 4d+69) then
        tmp = (y * z) / a
    else
        tmp = 1.0d0 * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -3.1e+17) {
		tmp = 1.0 * y;
	} else if (t <= 4e+69) {
		tmp = (y * z) / a;
	} else {
		tmp = 1.0 * y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= -3.1e+17:
		tmp = 1.0 * y
	elif t <= 4e+69:
		tmp = (y * z) / a
	else:
		tmp = 1.0 * y
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -3.1e+17)
		tmp = Float64(1.0 * y);
	elseif (t <= 4e+69)
		tmp = Float64(Float64(y * z) / a);
	else
		tmp = Float64(1.0 * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= -3.1e+17)
		tmp = 1.0 * y;
	elseif (t <= 4e+69)
		tmp = (y * z) / a;
	else
		tmp = 1.0 * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -3.1e+17], N[(1.0 * y), $MachinePrecision], If[LessEqual[t, 4e+69], N[(N[(y * z), $MachinePrecision] / a), $MachinePrecision], N[(1.0 * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < -3.1e17 or 4.0000000000000003e69 < t

   1. Initial program 67.7%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{y \cdot \left(z - t\right)}{a - t}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{y \cdot \left(z - t\right)}{\color{blue}{a - t}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{y \cdot \left(z - t\right)}{\color{blue}{a} - t} \]

      3. lower--.f64 N/A

         \[\leadsto \frac{y \cdot \left(z - t\right)}{a - t} \]

      4. lower--.f64 39.8

         \[\leadsto \frac{y \cdot \left(z - t\right)}{a - \color{blue}{t}} \]

   4. Applied rewrites 39.8%

      \[\leadsto \color{blue}{\frac{y \cdot \left(z - t\right)}{a - t}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{y \cdot \left(z - t\right)}{\color{blue}{a - t}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{y \cdot \left(z - t\right)}{\color{blue}{a} - t} \]

      3. associate-/l* N/A

         \[\leadsto y \cdot \color{blue}{\frac{z - t}{a - t}} \]

      4. lift--.f64 N/A

         \[\leadsto y \cdot \frac{z - t}{\color{blue}{a} - t} \]

      5. sub-negate-rev N/A

         \[\leadsto y \cdot \frac{\mathsf{neg}\left(\left(t - z\right)\right)}{\color{blue}{a} - t} \]

      6. lift--.f64 N/A

         \[\leadsto y \cdot \frac{\mathsf{neg}\left(\left(t - z\right)\right)}{a - t} \]

      7. lift--.f64 N/A

         \[\leadsto y \cdot \frac{\mathsf{neg}\left(\left(t - z\right)\right)}{a - \color{blue}{t}} \]

      8. sub-negate-rev N/A

         \[\leadsto y \cdot \frac{\mathsf{neg}\left(\left(t - z\right)\right)}{\mathsf{neg}\left(\left(t - a\right)\right)} \]

      9. lift--.f64 N/A

         \[\leadsto y \cdot \frac{\mathsf{neg}\left(\left(t - z\right)\right)}{\mathsf{neg}\left(\left(t - a\right)\right)} \]

      10. frac-2neg N/A

          \[\leadsto y \cdot \frac{t - z}{\color{blue}{t - a}} \]

      11. lift-/.f64 N/A

          \[\leadsto y \cdot \frac{t - z}{\color{blue}{t - a}} \]

      12. *-commutative N/A

          \[\leadsto \frac{t - z}{t - a} \cdot \color{blue}{y} \]

      13. lower-*.f64 52.1

          \[\leadsto \frac{t - z}{t - a} \cdot \color{blue}{y} \]

      14. lift-/.f64 N/A

          \[\leadsto \frac{t - z}{t - a} \cdot y \]

      15. frac-2neg N/A

          \[\leadsto \frac{\mathsf{neg}\left(\left(t - z\right)\right)}{\mathsf{neg}\left(\left(t - a\right)\right)} \cdot y \]

      16. lift--.f64 N/A

          \[\leadsto \frac{\mathsf{neg}\left(\left(t - z\right)\right)}{\mathsf{neg}\left(\left(t - a\right)\right)} \cdot y \]

      17. sub-negate-rev N/A

          \[\leadsto \frac{z - t}{\mathsf{neg}\left(\left(t - a\right)\right)} \cdot y \]

      18. lift--.f64 N/A

          \[\leadsto \frac{z - t}{\mathsf{neg}\left(\left(t - a\right)\right)} \cdot y \]

      19. lift--.f64 N/A

          \[\leadsto \frac{z - t}{\mathsf{neg}\left(\left(t - a\right)\right)} \cdot y \]

      20. sub-negate-rev N/A

          \[\leadsto \frac{z - t}{a - t} \cdot y \]

      21. lift--.f64 N/A

          \[\leadsto \frac{z - t}{a - t} \cdot y \]

      22. lower-/.f64 52.1

          \[\leadsto \frac{z - t}{a - t} \cdot y \]

   6. Applied rewrites 52.1%

      \[\leadsto \frac{z - t}{a - t} \cdot \color{blue}{y} \]

   7. Taylor expanded in t around inf

      \[\leadsto 1 \cdot y \]
   8. Step-by-step derivation

   9. Applied rewrites 25.4%

      \[\leadsto 1 \cdot y \]

   if -3.1e17 < t < 4.0000000000000003e69

   1. Initial program 67.7%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{y \cdot \left(z - t\right)}{a - t}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{y \cdot \left(z - t\right)}{\color{blue}{a - t}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{y \cdot \left(z - t\right)}{\color{blue}{a} - t} \]

      3. lower--.f64 N/A

         \[\leadsto \frac{y \cdot \left(z - t\right)}{a - t} \]

      4. lower--.f64 39.8

         \[\leadsto \frac{y \cdot \left(z - t\right)}{a - \color{blue}{t}} \]

   4. Applied rewrites 39.8%

      \[\leadsto \color{blue}{\frac{y \cdot \left(z - t\right)}{a - t}} \]

   5. Taylor expanded in t around 0

      \[\leadsto \frac{y \cdot z}{\color{blue}{a}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{y \cdot z}{a} \]

      2. lower-*.f64 16.7

         \[\leadsto \frac{y \cdot z}{a} \]

   7. Applied rewrites 16.7%

      \[\leadsto \frac{y \cdot z}{\color{blue}{a}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 25.4% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ 1 \cdot y \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 (* 1.0 y))

C

double code(double x, double y, double z, double t, double a) {
	return 1.0 * y;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = 1.0d0 * y
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return 1.0 * y;
}

Python

def code(x, y, z, t, a):
	return 1.0 * y

Julia

function code(x, y, z, t, a)
	return Float64(1.0 * y)
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = 1.0 * y;
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(1.0 * y), $MachinePrecision]

Derivation

1. Initial program 67.7%

   \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]

2. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{y \cdot \left(z - t\right)}{a - t}} \]
3. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{y \cdot \left(z - t\right)}{\color{blue}{a - t}} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{y \cdot \left(z - t\right)}{\color{blue}{a} - t} \]

   3. lower--.f64 N/A

      \[\leadsto \frac{y \cdot \left(z - t\right)}{a - t} \]

   4. lower--.f64 39.8

      \[\leadsto \frac{y \cdot \left(z - t\right)}{a - \color{blue}{t}} \]

4. Applied rewrites 39.8%

   \[\leadsto \color{blue}{\frac{y \cdot \left(z - t\right)}{a - t}} \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \frac{y \cdot \left(z - t\right)}{\color{blue}{a - t}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{y \cdot \left(z - t\right)}{\color{blue}{a} - t} \]

   3. associate-/l* N/A

      \[\leadsto y \cdot \color{blue}{\frac{z - t}{a - t}} \]

   4. lift--.f64 N/A

      \[\leadsto y \cdot \frac{z - t}{\color{blue}{a} - t} \]

   5. sub-negate-rev N/A

      \[\leadsto y \cdot \frac{\mathsf{neg}\left(\left(t - z\right)\right)}{\color{blue}{a} - t} \]

   6. lift--.f64 N/A

      \[\leadsto y \cdot \frac{\mathsf{neg}\left(\left(t - z\right)\right)}{a - t} \]

   7. lift--.f64 N/A

      \[\leadsto y \cdot \frac{\mathsf{neg}\left(\left(t - z\right)\right)}{a - \color{blue}{t}} \]

   8. sub-negate-rev N/A

      \[\leadsto y \cdot \frac{\mathsf{neg}\left(\left(t - z\right)\right)}{\mathsf{neg}\left(\left(t - a\right)\right)} \]

   9. lift--.f64 N/A

      \[\leadsto y \cdot \frac{\mathsf{neg}\left(\left(t - z\right)\right)}{\mathsf{neg}\left(\left(t - a\right)\right)} \]

   10. frac-2neg N/A

       \[\leadsto y \cdot \frac{t - z}{\color{blue}{t - a}} \]

   11. lift-/.f64 N/A

       \[\leadsto y \cdot \frac{t - z}{\color{blue}{t - a}} \]

   12. *-commutative N/A

       \[\leadsto \frac{t - z}{t - a} \cdot \color{blue}{y} \]

   13. lower-*.f64 52.1

       \[\leadsto \frac{t - z}{t - a} \cdot \color{blue}{y} \]

   14. lift-/.f64 N/A

       \[\leadsto \frac{t - z}{t - a} \cdot y \]

   15. frac-2neg N/A

       \[\leadsto \frac{\mathsf{neg}\left(\left(t - z\right)\right)}{\mathsf{neg}\left(\left(t - a\right)\right)} \cdot y \]

   16. lift--.f64 N/A

       \[\leadsto \frac{\mathsf{neg}\left(\left(t - z\right)\right)}{\mathsf{neg}\left(\left(t - a\right)\right)} \cdot y \]

   17. sub-negate-rev N/A

       \[\leadsto \frac{z - t}{\mathsf{neg}\left(\left(t - a\right)\right)} \cdot y \]

   18. lift--.f64 N/A

       \[\leadsto \frac{z - t}{\mathsf{neg}\left(\left(t - a\right)\right)} \cdot y \]

   19. lift--.f64 N/A

       \[\leadsto \frac{z - t}{\mathsf{neg}\left(\left(t - a\right)\right)} \cdot y \]

   20. sub-negate-rev N/A

       \[\leadsto \frac{z - t}{a - t} \cdot y \]

   21. lift--.f64 N/A

       \[\leadsto \frac{z - t}{a - t} \cdot y \]

   22. lower-/.f64 52.1

       \[\leadsto \frac{z - t}{a - t} \cdot y \]

6. Applied rewrites 52.1%

   \[\leadsto \frac{z - t}{a - t} \cdot \color{blue}{y} \]

7. Taylor expanded in t around inf

   \[\leadsto 1 \cdot y \]
8. Step-by-step derivation

9. Applied rewrites 25.4%

   \[\leadsto 1 \cdot y \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025154 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Chart.Axis.Types:linMap from Chart-1.5.3"
  :precision binary64
  (+ x (/ (* (- y x) (- z t)) (- a t))))